Here is the full list of all 16 Rocket Knight achievements.
Completed Arcade Mode on Normal
Completed all stages
Complete World 4
Complete World 3
Complete World 2
Complete World 1
Completed Arcade Mode on Hard
Completed Gold Sparkster Mode
Completed all stages below par time
Completed Arcade Mode scoring 1,000,000 or more
Completed Arcade Mode on Hard within 60 minutes
Completed 1.2 having defeated all enemies
Completed 2.1 having collected all pickups
Completed 3.2 without taking damage
Defeat 5 enemies in a single Rocket Burst or Drill attack
Defeat 3 enemies with a single reflected projectile